nine astronomical elements,* to which the tables of India assign such values as do by no means belong to them in these later ages, but such as the theory of gravity proves to have belonged to them three thousand years before the Christian æra. At that time, therefore, or in the ages preceding it, the observations must have been made from which these elements were deduced. For it is abundantly evident, that the Brahmins of later times, however willing they might be to adapt their tables to so remarkable an epoch as the Kaly-Yug, could never think of doing so, by substituting, instead of quantities which they had observed, others which they had no reason to believe had ever existed. The elements in question are precisely what these astronomers must have sup

*"The inequality or the precession of the equinoxes; the acceleration of the moon; the length of the solar year; the equation of the sun's centre; the obliquity of the ecliptic; the place of Jupiter's aphelion; the equation of Saturn's centre; and the inequalities in the mean motion of both these planets."

posed invariable, and of which, had they supposed them to change, they had no rules to guide them for ascertaining the variations; since to the discovery of these rules is required, not only all the perfection to which astronomy is at this day brought in Europe, but all that which the sciences of motion and of extension have likewise attained. It is equally clear that these coincidences are not the work of accident; for it will scarcely be supposed that chance has adjusted the errors of the Indian astronomy with such singular felicity, that observers, who could not discover the true state of the heavens, at the age in which they lived, have succeeded in describing one which took place several thousand years before they were born.*

The preceding calculations must have required the assistance of many subsidiary tables, of which no trace has yet been

* See Trans. of the Royal Society of Edinburgh, vol. ii. pp. 169, 170.

found in India,-besides many other geometrical propositions. Some of them also involve the ratio which the diameter of a circle was supposed to bear to its circumference, but which we should find it impossible to discover from them exactly, on account of the small quantities that may have been neglected in their calculations. Fortunately, we can arrive at this knowledge, which is very material when the progress of geometry is to be estimated, from a passage in the Ayin Akbery, where we are told that the Hindus suppose the diameter of a circle to be to its circumference, as 1250 to 3927; and where the author, who believed it to be perfectly exact, expresses his astonishment, that among so simple a people, there should be found a truth, which among the wisest and most learned nations had been sought for in vain.

The proportion of 1250 to 3927, is indeed a nearer approach to the quadrature of the circle; it differs little from that

of Metius,* 113 to 355, and is the same with one equally well known, that of 1 to 3.1416. When found in the simplest and most elementary way, it requires a polygon of 768 sides to be inscribed in a circle; an operation which cannot be arithmetically performed without the knowledge of some very curious properties of that curve, and at least nine extractions of the square root, each as far as ten places of decimals. All this must have been accomplished in India; for, it is to be observed, that the above-mentioned proportion cannot have been received from the mathematicians of the west. The Greeks left nothing on this subject more accurate than the theory of Archimedes; and the Arabian mathematicians seem not to have attempted any nearer approximation. The geometry of modern Europe can much less be regarded as the source of this knowledge. Metius

* Adrian Metius, native of Alkmaar, in Holland. The discovery of spying glasses is attributed to his brother, James Metius.

and Vieta* were the first, who, in the quadrature of the circle, surpassed the accuracy of Archimedes; they flourished at the very time when the Institutes of Akber were collected in India.†

"On the grounds which have now been explained, the following general conclusions appear to be established.

"1st. The observations on which the astronomy of India is founded, were made more than three thousand years before the Christian æra; and, in particular, the places of the sun and moon, at the beginning of the Kaly-Yug, were determined by actual observation.

This follows from the exact argument of the radical places in the tables of Tirvalore, with those deduced for the same epoch from the tables of De la Caille and Mayer, and especially in the case of the

* Francis Vieta was a native of Fontenai in Poitou. He was born in 1540, and died in 1603.

+ See Trans. of the Royal Society of Edinburgh, vol. ii. p. 185.